The $K_4^-$-density conjecture in $C_5$-free $3$-graphs

Let $C_5$ be a $3$-graph and let $K_4^-$ be the complete $3$-graph on four vertices with one edge removed. For a forbidden $3$-graph $F$, write $\pi_{K_4^-}(F)$ for the maximum asymptotic induced density of $K_4^-$ in an $F$-free $3$-graph.

The $K_4^-$-density conjecture.

$$\pi_{K_4^-}(C_5)=4-6\left((\sqrt{2}+1)^{1/3}-(\sqrt{2}-1)^{1/3}\right).$$

This is the lower bound supplied by the iterated unbalanced blow-up construction; the source records only numerical upper and lower bounds, so equality remains open.